Theorem fsumclf

Description: Closure of a finite sum of complex numbers A ( k ) . A version of fsumcl using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fsumclf.ph	|- F/ k ph
	fsumclf.a	|- ( ph -> A e. Fin )
	fsumclf.b	|- ( ( ph /\ k e. A ) -> B e. CC )
Assertion	fsumclf	|- ( ph -> sum_ k e. A B e. CC )

Proof

Step	Hyp	Ref	Expression
1		fsumclf.ph	|- F/ k ph
2		fsumclf.a	|- ( ph -> A e. Fin )
3		fsumclf.b	|- ( ( ph /\ k e. A ) -> B e. CC )
4		csbeq1a	|- ( k = j -> B = [_ j / k ]_ B )
5		nfcv	|- F/_ j B
6		nfcsb1v	|- F/_ k [_ j / k ]_ B
7	4 5 6	cbvsum	|- sum_ k e. A B = sum_ j e. A [_ j / k ]_ B
8	7	a1i	|- ( ph -> sum_ k e. A B = sum_ j e. A [_ j / k ]_ B )
9		nfv	|- F/ k j e. A
10	1 9	nfan	|- F/ k ( ph /\ j e. A )
11	6	nfel1	|- F/ k [_ j / k ]_ B e. CC
12	10 11	nfim	|- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
13		eleq1w	|- ( k = j -> ( k e. A <-> j e. A ) )
14	13	anbi2d	|- ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
15	4	eleq1d	|- ( k = j -> ( B e. CC <-> [_ j / k ]_ B e. CC ) )
16	14 15	imbi12d	|- ( k = j -> ( ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC ) ) )
17	12 16 3	chvarfv	|- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
18	2 17	fsumcl	|- ( ph -> sum_ j e. A [_ j / k ]_ B e. CC )
19	8 18	eqeltrd	|- ( ph -> sum_ k e. A B e. CC )